viscomp

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viscomp

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viscomp

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Index

Shaders Traslation, rotation, scaling and sheared --> 2d, nonlinear,.. Homogeneous space Traslation, rotation(Euler), scaling and sheared --> 3d, linear,.. Orthogonal Matrix Rotations Quaternion Rotations Modeling Transformations

PIPELINE

Shaders In Processing

uniform mat4 transform;
attribute vec4 vertex;
attribute vec4 color;
varying vec4 vertColor;

void main(){
   gl_Position = transform * vertex;
   vertColor = color;
}
varying vec4 vertColor;

void main(){
  gl_FragColor = vertColor;
}

Two-dimensional transformations

Two-dimensional transformations

Object Transformations vs Coordinate System Transformation

Two-dimensional transformations

Active Transformation (Standard Basis) vs Passive Transformation (change of base)

Two-dimensional transformations

Traslation

Vectorially we get:

[xy][dxdy][x′y′]

P′=P+T

Two-dimensional transformations

Scaling

Vectorially we get:

[xy][sx00sy][x′y′]

P′=S∗P

Two-dimensional transformations

Rotation

Two-dimensional transformations

Vectorially we get:

[xy][cosβ−sinβsinβcosβ][x′y′]

P′=R∗P

Rotation

From the triangulation:

x=rcosα

y=rsinα

x′=rcos(α+β)=rcosαcosβ−rsinαsinβ

y′=rsin(α+β)=rcosαsinβ−rsinαcosβ

x′=xcosβ−ysinβ

y′=xsinβ+ycosβ

Two-dimensional transformations

Rotation

We have

x′=x+h∗y,y′=y

,where h is the sheared parameter, to obtain:

Dx=[1h01]Dy=[10h1]P′=DxDy∗P

Two-dimensional transformations

T=[dxdy]

S=[sx00sy]

R=[cosβ−sinβsinβcosβ]

Dx=[1h01]Dy=[10h1]

Trasformations Summary

Traslation

P′=P+T

Scaling

P′=S∗P

Rotation

P′=R∗P

Sheared

P′=DxDy∗P

Types of Transformations

1. Linear

F(a+b)=F(a)+F(b)F(λa)=λF(a)→F(0)=0 , Thus is a nonlinear transformation

Note: Matrix multiplication is always linear

2. Related

Linear + Traslation →p′=M∗p+b

3. Invertibles

Lets take a look at M−1, the inverse matrix of the general transformation matrix M

P′=M∗P,       M−1P′=M−1MP

Homogeneous space

Homogeneous coordinates

(x,y,w)

2D Space → Homogeneous Space

(x,y)→(x,y,1)

(xw,yw)→(x,y,w)

3D Space → Homogeneous Space

(x,y,z)→(x,y,z,1)

(xw,yw,zw)→(x,y,z,w)

Summary

T=[10dx01dy001]

S=[sx000sy0001]

R=[cosβ−sinβ0sinβcosβ0001]

Dx=[1h0010001]Dy=[100h10001]

P′=[x′y′1]P=[xy1]

Traslation

P′=T(dx,dy)∗P

Scaling

P′=T(sx,sy)∗P

Rotation

P′=R(β)∗P

Sheared

P′=DxDy∗P

Homogeneous coordinates

Observation about the inverse matrix of the transformation matrix

Lets take a look at M−1, the inverse matrix of the general transformation matrix M

P′=M∗P,       M−1P′=M−1MP

M−1P′=I   P=P

T=[10dx01dy001]T−1=[10−dx01−dy001]

S=[sx000sy0001]S−1=[1sx0001sy0001]

P′=[x′y′1]P=[xy1]

Traslation

P′=T(dx,dy)∗P

Scaling

P′=T(sx,sy)∗P

Homogeneous coordinates

Observation about the inverse matrix of the transformation matrix

Rotation

P′=R(β)∗P      R=[cosβ−sinβ0sinβcosβ0001] R=[cos(−β)−sin(−β)0sin(−β)cos(−β)0001]=[cosβsinβ0−sinβcosβ0001]

Two-dimensional transformations Composition

Successive transformations example

P
→
P′

Traslation
→
Rotation
→
Traslation
→
Scaling

P1=T∗P
→
P2=R∗P1
→
P3=T′∗P2
→
P′=S∗P3

P′=S∗T′∗R∗T∗P

Two-dimensional transformations Composition

Some special compositions

Rotation from pivot point

Two-dimensional transformations Composition

Some special compositions

Rotation from pivot point

Two-dimensional transformations Composition

Some special compositions

Rotation from pivot point

[10xr01yr001] ∙ [cosβ−sinβ0sinβcosβ0001] ∙ [10−xr01−yr001]

=[cosβ−sinβxr(1−cosβ)+yrsinβsinβcosβyr(1−cosβ)−xrsinβ001]

Two-dimensional transformations Composition

Some special compositions

Scaling from static point

Two-dimensional transformations Composition

Some special compositions

Scaling from static point

Two-dimensional transformations Composition

Some special compositions

Scaling from static point

[10xf01yf001] ∙ [sx000sy0001] ∙ [10−xf01−yf001]

=[sx0xf(1−sx)0syyf(1−sy)001]

Two-dimensional transformations Composition

Some special compositions

General directions to scaling

1. The object is rotated to angle β, so the directions s1 y s2 match with the axes.

2. Scale the object

3. The object is rotated on the opposite way.

Two-dimensional transformations Composition

Some special compositions

General directions to scaling

[s1cos2β+s2sin2β(s2−s1)cosβsinβ0(s2−s1)cosβsinβs1sin2β+s2cos2β0001]

Traslations

Traslations

Specification

Vectorially we get:

[xyz] [dxdydz] [x′y′z′]

P′= P + T

In homogeneous coordinates we get:

[xyz1] [100dx010dy001dz0001] [x′y′z′1]

P′= T∗ P

Scaling

Scaling

Specification

Vectorially we get:

[xyz] [sx000sy000sz] [x′y′z′]

P′= S∗ P

In homogeneous coordinates we get:

[xyz1] [sx0000sy0000sz00001] [x′y′z′1]

P′= S∗ P

Rotations

Parameters used in rotations

Rotations

Rotation respect z-axis

Notice z=z′, we get:

x′=xcosβ−ysinβ

y′=xsinβ+ycosβ

z′=z

In homogeneous coordinates we get:

[xyz1] [cosβ−sinβ00sinβcosβ0000100001] [x′y′z′1]

P′=R∗P

Rotations

Rotation respect x-axis

Doing y→z→x→y

In homogeneous coordinates we get:

[xyz1] [10000cosβ−sinβ00sinβcosβ00001] [x′y′z′1]

P′=R∗P

Rotations

Rotation respect y-axis

Doing y→z→x→y

In homogeneous coordinates we get:

[xyz1] [cosβ0senβ00100−sinβ0cosβ00001] [x′y′z′1]

P′=R∗P

Rotations

General Rotations

Rotations

Rotations

1st step. Rotation parameter specifications

V=P2−P1

u=V|V|=(a,b,c)

a=(x2−x1)|V|

b=(y2−y1)|V|

a=(z2−z1)|V|

Rotations

2nd step. Translation from P1 to origin

T=[100−x1010−y1001−z10001]

Rotations

3rd step. Turn of P2 in z-axis

Rotations

3rd step

Rx(α)=[10000cd−bd00bdcd00001]

We get:

u= (a,b,c). From u projection in y-z plane:

u′= (0,b,c). α is the angle between u′ y uz→

cosα=(u′∙uz)(|u′||uz|)=cd

whered=√b2+c2=|u′|,

Finally from cross product:

1.u′xuz=ux|u′||uz|sinα (Idempotent form) y

2.u′xuz=uxb (cartesian from)

Thus:

sinα=bd

Rotations

3rd step

Ry(λ)=[d0−a00100a0d00001]

We get:

u= (a,b,c). From u projection in y-z plane:

u′= (0,b,c).

From u rotation respect x-axis we get:

u″= (a,0,d )with d=√b2+c2 (u′ magnitude)

λ is the angle between u″ y uz→

cosλ=(u″∙uz)(|u″||uz|)=d,since|u″|=|uz|=1

Finally from cross product:

1.u″xuz=uy|u″||uz|sinλ (Idempotent form) y

2.u″xuz=uy(−a) (cartesian from)

Thus: sinλ=−a

Rotations

Rz(β)=[cosβ−sinβ00sinβcosβ0000100001]Rx(α)−1∙Ry(λ)−1T−1

R(β)=T−1∙Rx(α)−1∙Ry(λ)−1∙Rz(β)∙Ry(λ)∙Rx(α)∙T

Rotations

Another way to calculate the composite transformation matrix:

Ry(λ)∙Rx(α)

Rotations

Another way to calculate the composite transformation matrix:

Ry(λ)∙Rx(α)

V=P2−P1

u=V|V|=(a,b,c)

a=(x2−x1)|V|

b=(y2−y1)|V|

c=(z2−z1)|V|

Rotations

Another way to calculate the composite transformation matrix:

Ry(λ)∙Rx(α)

u=V|V|=(a,b,c)

u′z=u(z′-axis aligns with z)

u′x=any unit vector(in the orthogonal plane to u′z)

u′y=u x u′x

u′x=(u′x1,u′x2,u′x3)

u′y=(u′y1,u′y2,u′y3)

u′x=(u′z1,u′z2,u′z3)

R=[u′x1u′x2u′x30u′y1u′y2u′y30u′z1u′z1u′z100001]

Rotations

Rotations and Quaternions - Review

Concept:  q=s+ia+jb+kc

The coefficients a,b and c, in the imaginary terms i,j and k, are real.

S parameter is a real number known as scalar part.

i2=j2=k2=−1    ij=−ji=k(Definition)

jk=−kj=i         ki=−ik=j(Derived)

Sum (q1,q2):  q1,q2=(s1+s2)+i(a1+a2)+j(b1+b2)+k(c1+c2)

Scalar Multiplication (d):  dq=(ds)+i(da)+j(db)+k(dc)

Rotations

Rotations and Quaternions - Review

Vectorial Notation:  q=(s,v)  where v=(a,b,c)

Product(q1,q2):  q1q2=(s1s2−v1∙v2,s1v2+s2v1+v1xv2)

|q|2=s2+v∙v

q−1=(1|q|2)(s,−v)  note:(s,−v) is called the"conjugate"of q

thus: qq−1=q−1q=(1,0)

Rotations

The hypersphere of rotations(2D case)

Rotation angle : β(->latitude α)

Observations

1. North and south poles -> identity rotation. No axis is defined

2. Each rotation can be represented as a rotation about some axis, or, equivalently, as a negative rotation about an axis pointing in the opposite direction: antipodal point.

3. The set of rotations is not closed under composition.

Rotations

The hypersphere of rotations(3D case)

Rotation angle : β(->latitude α)

Observations

1. Hypersphere in four dimensional space.

2. The "latitude" on the hypersphere will be half of the corresponding angle of rotation.

3. The set of rotations is closed under composition.

4. A general quaternion represents a point in 4D, but constraining it to have unit magnitude yields a three dimensional space equivalent to the surface of a hypersphere.

Rotations

Parameterizing the space of rotations(2D case)

Problem

1. Latitude and longitude are degenerated.

2. It can be shown that no two-parameter coordinate system can avoid such degeneracy.

Idea

Parameterize the space with three Cartesian coordinates (here w,x,y)

North pole -> (w,x,y)=(1,0,0) South pole -> (w,x,y)=(−1,0,0) equator -> w=0,x2+y2=1

Points on the sphere satisfy the constraint: w2+x2+y2=1.

A point (w,x,y) on the sphere represents a rotation

Horizontal axis directed by the vector (x,y,0).

* Angle β=2∗cos−1w

1.w=cosα,sinceα=β2→w=cos(β/2)

2. Since w2=cos2(β/2)→(x2+y2=sin2(β/2))

Rotations

Parameterizing the space of rotations(3D case)

Problem

1. Latitude and longitude are degenerated.

2. It can be shown that no two-parameter coordinate system can avoid such degeneracy.

Idea

Parameterize the space with four Cartesian coordinates (here w,x,y,z)

Points on the hypersphere satisfy the constraint:w2+x2+y2+z2=1

A point (w,x,y,z) on the sphere represents a rotation

\ast Axis directed by the vector (x,y,z).

\ast Angle β=2∗cos−1w

1.w=cosα,sinceα=β2→w=cos(β/2)

2. Since w2=cos2(β/2)→(x2+y2+z2=sin2(β/2))

Rotations

Rotations and Quaternions

u is a unit vector

β is the rotation angle

For the rotation around an axis across the origin

1.Defined the unit quaternion:

q=(s,v) where s=cos(β/2)

v=usin(β/2)

The position of any point P(to turn through this quaternion), it can be represented as:P=(0,p)

With p=(x,y,z) (coordinates), we get:P=(0,p)

2.P′=qPq−1, the rotation point is defined.

With q−1=(s,−v) this is the inverse of q

Rotations

Proof and Properties

Let ube a unit vector (rotation axis) and q=(s, uv=cos(β/2)+ usin(β/2).The goal is to show that:

(1)P′=qPq−1→ yields the vector P′ upon rotation of the original vector P by an angleβ around the axis u.

Proof: expand (1) to reduce it to rodrigues rotation formula. See next slide.

Important properties

1. Quaternion multiplication is composition of rotations, for if p and q are quaternions representing rotations, then rotation (conjugation) by pq is: pqP(pq)−1 =pqPq−1p−1 = p(qPq−1)p−1 which is the same as rotating (conjugating) by q and then by p.

2. The quaternion inverse of a rotation is the opposite rotation, since q−1(qPq−1)q=P

3.qn is a rotation by n times the angle around the same axis as q.

4. This (3) can be extended to arbitrary real n, allowing for smooth interpolation between spatial orientations.

Rotations

Rodrigues rotation formula

The following formula rotates v around the unit vectorz by an angle β:

vrot=vcosβ+(zxv)sin\beta+z(z\bulletv)(1-cos\beta)

Observation

Let \phi be the angle between z and v.

z\bullet v =|z||v|cos\phi(1),

|zx v |=|z||v|sin\phi(2)

Since|z|=1 \rightarrow

a)|y|=|v|sin\phi(from(2))

b)z\bullet v =|(z\bulletv)z|=|v| cos\phi

Conclusion: |x|=|y|

x_{rot}=xcos\beta +ysin\beta

x_{rot}=(v-(z\bulletv)z) cos\beta +(zxv) sin\beta

v_{rot}=xrot+(z\bulletv)z

Rotations

Rotations and Quaternions

Consider the traslation:

R(\beta)=T^{-1}\bullet M_R(\beta) \bulletT

We got before:

u is a unit vector

\beta is the rotation angle

q =(s,v) ,with s= cos(\beta/2)

v=usin(\beta/2)

q^{-1}=(s,-v)\space \space p= (x,y,z)

P'=qPq^{-1}=(0,p')

p'= s^2p+v(p\bulletv) + 2s(vxp)+vx(vxp)

Taking v= (a,b,c)we get:

M_R(\beta)=\begin{bmatrix} 1-2b^2-2c^2 & 2ab-2sc & 2ac+2sb & 0 \cr 2ab+2sc & 1-2a^2-2c^2 & 2bc-2sa & 0 \cr 2ac-2sb & 2bc+2sa & 1-sa^2-2b^2 & 0 \cr 0 & 0 & 0 & 1 \end{bmatrix}

R(\beta)=T^{-1}\bullet R_x(\alpha)^{-1} \bullet R_y(\lambda)^{-1} \bullet R_z(\beta) \bullet R_y (\lambda) \bullet R_x(\alpha) \bulletT

Shear

Three-dimensional/ respect to z-axis

D_z=\begin{bmatrix} 1 & 0 & a & 0 \cr 0 & 1 & b & 0 \cr 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 \end{bmatrix} \space \space \space

Thus:

x'= x +a\ast z

y'= y +b\ast z

z'= z

Modeling Transformations

World and local coordinates systems

Modeling Transformations

Transformations between coordinates systems

Note:Transformations between systems may involve till three consecutive rotations

Modeling Transformations

Transformations between coordinates systems/ Hierarchy(Graph)

Modeling Transformations

Transformations between coordinates systems/ Orthogonal Matrix method

Coordinate transformation involve 1 traslation and 3 rotations: \space R \bullet T

Alternative way to calculate the composed rotation matrix: R

u_x'=(u_x'1,u_x'2,u_x'3)

u_y'=(u_y'1,u_y'2,u_y'3)

u_x'=(u_z'1,u_z'2,u_z'3)

R=\begin{bmatrix} u_x'1 & u_x'2 & u_x'3 & 0 \cr u_y'1 & u_y'2 & u_y'3 & 0 \cr u_z'1 & u_z'1 & u_z'1 & 0 \cr 0 & 0 & 0 & 1 \end{bmatrix}

Modeling Transformations

OpenGL Example

R_1T_1:S_1\rightarrowS_wtranslate and rotateS_w\rightarrowS_1

R_2T_2:S_2\rightarrowS_1translate and rotateS_1\rightarrowS_2

R_3T_3:S_3\rightarrowS_1translate and rotateS_3\rightarrowS_1

R_4T_4:S_4\rightarrowS_2translate and rotateS_2\rightarrowS_4

R_cT_c:S_c\rightarrowS_wtranslate and rotateS_w\rightarrowS_c

Modeling Transformations

OpenGL Example

OpenGL matrix stack (model-view)

pushMatrix: multiply current matrix and adds to the head

popMatrix: remove head